The symplectic representation of the mapping class group is unique

The mapping class group, MCG, of a closed orientable surface  of
genus g has classical symplectic representation coming from its  action on
the first
homology of the surface. I will show that any 2g-dimensional  nontrivial
representation of the group MCG is conjugate to this  classical
representation. I will also prove that MCG has no faithful  linear
representation in dimensions less than 3g-2. This contributes  to the
well-known problem on the linearity of MCG. Some algebraic  corollaries on
representations of related group will be given.